A counterexample in nonlinear interpolation

Abstract:

If $({A_0},{A_1})$ is an interpolation pair with ${A_0} \subset {A_1}$ and T is a possibly nonlinear operator which maps ${A_0}$ into ${A_0}$ and ${A_1}$ into ${A_1}$ and satisfies ${\left \| {Ta} \right \|_{{A_0}}} \leqslant C{\left \| a \right \|_{{A_0}}}$ and ${\left \| {Tb - Tb’} \right \|_{{A_1}}} \leqslant C{\left \| {b - b’} \right \|_{{A_1}}}$ for all $a \in {A_0}$ and b, $b,b’ \in {A_1}$ and for some constant C, then it is known that T also maps the real interpolation spaces ${({A_0},{A_1})_{\theta ,p}}$ into themselves. We give an example showing that T need not map the complex interpolation spaces ${[{A_0},{A_1}]_\theta }$ into themselves. It is also seen that quasilinear operators may fail to preserve complex interpolation spaces.